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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 4

On topological homotopy groups and relation to Hawaiian groups

Ameneh Babaee(Ferdowsi University of Mashhad)0000-0001-5243-0641
Behrooz Mashayekhy(Ferdowsi University of Mashhad)0000-0001-5243-0641
Hanieh Mirebrahimi(Ferdowsi University of Mashhad)0000-0001-5243-0641
Hamid Torabi(Ferdowsi University of Mashhad)0000-0001-5243-0641
Mahdi Abdullahi Rashid(Ferdowsi University of Mashhad)0000-0001-5243-0641
Seyyed Zeynal Pashaei(Ferdowsi University of Mashhad)0000-0001-5243-0641
Sayfa
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Yayın Tarihi
2020-08-06
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Finally, we show that $n$-SLT paths can transfer $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $\pi_n^{wh}(X, x_0)$, we show that $\pi_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$.
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Temel Bulgular & Veri:Also, we present some necessary and sufficient conditions for $\pi_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete.
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Sonuç & Katkı:Finally, we show that $n$-SLT paths can transfer $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Whisker topology · Hawaiian group · Harmonic archipelago

Abstract

By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $\pi_n^{wh}(X, x_0)$, we show that $\pi_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $\pi_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $\pi_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $\pi_n^{wh}(X, x_0) \cong \pi_n^{wh}(Y, y_0)$, but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally $n$-simply connected spaces and $n$-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of $n$-loop space coincide. Finally, we show that $n$-SLT paths can transfer $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.

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Bu Makaleye Atıf Yap

Babaee, A., Mashayekhy, B., Mirebrahimi, H., Torabi, H., Rashid, M. A., Pashaei, S. Z. (2020). On topological homotopy groups and relation to Hawaiian groups. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 4. https://doi.org/10.15672/hujms.565367

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